Fresh from their 3-0 victory over the Gunners last weekend, Roberto Martinez's side had to wait until the 75th minute to score at the Stadium of Light when former Manchester United defender Wes Brown put Gerard Deulofeu's cross into his own net.
Victory moved Everton in the Champions League places, two points ahead of Arsenal, who were in FA Cup semi-final action against Wigan at Wembley, with both clubs having five league games remaining this season.
All major matches in England kicked off seven minutes late as a tribute to the looming 25th anniversary of the Hillsborough disaster that killed 96 Liverpool fans.
At the Hawthorns, West Brom squandered a 3-0 lead in a 3-3 draw with Tottenham Hotspur.
Matej Vydra gave West Brom a first-minute lead from close range when Spurs and France goalkeeper Hugo Lloris palmed a Morgan Amalfitano cross into his path.
Things got even better for the Baggies three minutes later when Chris Brunt volleyed home after Spurs failed to clear a Steven Reid cross.
Travelling Spurs fans must have feared the north London club's luck was out when, after Amalfitano brought down Danny Rose, Emmanuel Adebayor's weak penalty was saved by England goalkeeper Ben Foster.
West Brom then made it 3-0 in the 31st minute when a poor backwards header by Spurs' Younes Kaboul allowed Stephane Sessegnon to run through and score.
Spurs, however, did pull a goal back three minutes later when Jonas Olsson deflected Aaron Lennon's ball across the box into his own net.
The visitors made it 3-2 with 20 minutes left when Harry Kane headed in a Lennon cross and sixth-placed Spurs' comeback was complete in stoppage time when Christian Eriksen smashed in an equaliser.
Meanwhile things tightened up at the foot of the table as relegation-threatened Fulham and second-bottom Cardiff won to close the gap on the teams immediately above them.
Norwich may have sacked manager Chris Hughton in a bid to preserve their Premier League status but the Canaries first match under former youth coach Neil Adams left them even closer to relegation after a 1-0 defeat at Fulham.
The Canaries fell behind when Hugo Rodallega rifled the ball into the roof of a net five minutes before half-time at Craven Cottage.
Victory left Fulham still in the bottom three but saw the west London club close the gap between themselves and Norwich to just two points with four games remaining.
Cardiff meanwhile moved to within three points of safety with a shock 1-0 win away to Southampton, with Juan Cala's 65th-minute shot from the edge of the box proving decisive at St Mary's.
Crystal Palace moved closer to survival with a 1-0 win away to Aston Villa that left the Birmingham club just four points above the relegation zone.
Jason Puncheon's 76th-minute goal was all that separated the sides at Villa Park.
Elsewhere, Stoke beat Newcastle 1-0 when Erik Pieters's 42nd minute cross floated over the head of Magpies keeper Tim Krul.
Sunday could have a key bearing on the destiny of the Premier League title when leaders Liverpool face third-placed Manchester City and Chelsea, currently second, travel to Swansea.
Liverpool will kick-off four points in front of City, and two ahead of Chelsea, at what is sure to be a highly-charged Anfield.
City, however, have two games in hand on both their title rivals.